In earlier sections we studied linear transformations from to . In this chapter we will define and study linear transformations from a general vector space to a general vector space . The results we will obtain here have important applications in physics, engineering, and various branches of mathematics.
8.1 General Linear Transformations
Up to now our study of linear transformations has focused on transformations from to . In this section we will turn our attention to linear transformations involving general vector spaces. We will illustrate ways in which such transformations arise, and we will establish a fundamental relationship between general -dimensional vector spaces and .
Definitions and Terminology
In Section 1.8 we defined a matrix transformation to be a mapping of the form
in which is an matrix. We subsequently established in Theorem 1.8.3 that the matrix transformations are precisely the linear transformations from to , that is, the transformations with the linearity properties
We will use these two properties as the starting point for defining more general linear transformations. ...